The problem asks us to identify which ordered pairs satisfy the given linear equation: $y = -\frac{5}{6}x + \frac{2}{3}$.

1. Problem Description

The problem asks us to identify which ordered pairs satisfy the given linear equation: $y = -\frac{5}{6}x + \frac{2}{3}$.

2. Solution Steps

The equation given is $y = -\frac{5}{6}x + \frac{2}{3}$. To check if a point $(x, y)$ lies on the graph of the equation, we substitute the $x$ and $y$ values into the equation and see if the equation holds true.

* $(4, 7)$: $7 = -\frac{5}{6}(4) + \frac{2}{3} = -\frac{20}{6} + \frac{4}{6} = -\frac{16}{6} = -\frac{8}{3}$. $7 \ne -\frac{8}{3}$, so $(4, 7)$ is not on the graph.

* $(3, -3)$: $-3 = -\frac{5}{6}(3) + \frac{2}{3} = -\frac{15}{6} + \frac{4}{6} = -\frac{11}{6}$. $-3 \ne -\frac{11}{6}$, so $(3, -3)$ is not on the graph.

* $(2, -1)$: $-1 = -\frac{5}{6}(2) + \frac{2}{3} = -\frac{10}{6} + \frac{4}{6} = -\frac{6}{6} = -1$. $-1 = -1$, so $(2, -1)$ is on the graph.

* $(-4, 4)$: $4 = -\frac{5}{6}(-4) + \frac{2}{3} = \frac{20}{6} + \frac{4}{6} = \frac{24}{6} = 4$. $4 = 4$, so $(-4, 4)$ is on the graph.

* $(-6, -2)$: $-2 = -\frac{5}{6}(-6) + \frac{2}{3} = 5 + \frac{2}{3} = \frac{15}{3} + \frac{2}{3} = \frac{17}{3}$. $-2 \ne \frac{17}{3}$, so $(-6, -2)$ is not on the graph.

* $(-3, -5)$: $-5 = -\frac{5}{6}(-3) + \frac{2}{3} = \frac{15}{6} + \frac{4}{6} = \frac{19}{6}$. $-5 \ne \frac{19}{6}$, so $(-3, -5)$ is not on the graph.

3. Final Answer

(2, -1), (-4, 4)